Antarctic Whale Project: Single Species
Ben Weinstein
2016-09-02 13:37:09
1 Descriptive Statistics
1.1 By Month
1.2 Distance
1.3 Time
1.4 Velocity
1.5 Angles
2 Correlated random walk
Process Model
\[ d_{t} \sim T*d_{t-1} + Normal(0,\Sigma)\] \[ x_t = x_{t-1} + d_{t} \]
2.1 Parameters
For each individual:
\[\theta = \text{Mean turning angle}\] \[\gamma = \text{Move persistence} \]
For both behaviors process variance is: \[ \sigma_{latitude} = 0.1\] \[ \sigma_{longitude} = 0.1\]
2.2 Behavioral States
\[ \text{For each individual i}\] \[ Behavior_1 = \text{traveling}\] \[ Behavior_2 = \text{foraging}\]
\[ \alpha_{i,1,1} = \text{Probability of remaining traveling when traveling}\] \[\alpha_{i,2,1} = \text{Probability of switching from Foraging to traveling}\]
\[\begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix} \]
2.3 Environment
Behavioral states are a function of local environmental conditions. The first environmental condition is ocean depth. I then build a function for preferential foraging in shallow waters.
It generally follows the form, conditional on behavior at t -1:
\[Behavior_t \sim Multinomial([\phi_{traveling},\phi_{foraging}])\]
With the probability of switching states:
\[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}} + \beta_{Month,1} * Ocean_{y[t,]} + \beta_{Month,2} * Coast_{y[t,]}\]
\[logit(\phi_{foraging}) = \alpha_{Behavior_{t-1}} \]
Following Bestley in preferring to describe the switch into feeding, but no estimating the resumption of traveling.
The effect of the environment is temporally variable such that
\[ \beta_{Month,2} \sim ~ Normal(\beta_{\mu},\beta_\tau)\]
2.4 Continious tracks
The transmitter will often go dark for 10 to 12 hours, due to weather, right in the middle of an otherwise good track. The model requires regular intervals to estimate the turning angles and temporal autocorrelation. As a track hits one of these walls, call it the end of a track, and begin a new track once the weather improves. We can remove any micro-tracks that are less than three days. Specify a duration, calculate the number of tracks and the number of removed points. Iteratively.
How did the filter change the extent of tracks?
sink(“Bayesian/Multi_RW.jags”) cat(" model{
#Constants
pi <- 3.141592653589
##argos observation error##
argos_prec[1:2,1:2] <- inverse(argos_sigma*argos_cov[,])
#Constructing the covariance matrix
argos_cov[1,1] <- 1
argos_cov[1,2] <- sqrt(argos_alpha) * rho
argos_cov[2,1] <- sqrt(argos_alpha) * rho
argos_cov[2,2] <- argos_alpha
for(i in 1:ind){
for(g in 1:tracks[i]){
## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec)
#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)
###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs
#Process Model for movement
for(t in 2:(steps[i,g]-1)){
#Behavioral State at time T
logit(phi[i,g,t,1]) <- alpha_mu[state[i,g,t-1]] + beta[Month[i,g,t-1],state[i,g,t-1]] * ocean[i,g,t] + beta2[Month[i,g,t-1],state[i,g,t-1]] * coast[i,g,t]
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])
#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])
#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])
#Gaussian Displacement
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
}
#Final behavior state
logit(phi[i,g,steps[i,g],1]) <- alpha_mu[state[i,g,steps[i,g]-1]] + beta[Month[i,g,steps[i,g]-1],state[i,g,steps[i,g]-1]] * ocean[i,g,steps[i,g]] + beta2[Month[i,g,steps[i,g]-1],state[i,g,steps[i,g]-1]] * coast[i,g,steps[i,g]]
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])
## Measurement equation - irregular observations
# loops over regular time intervals (t)
for(t in 2:steps[i,g]){
# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]
#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec)
}
}
}
}
###Priors###
#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)
##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)
# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi
# prior for theta in 'foraging state'
theta[2] <- (tmp[2] * pi * 2)
##Move persistance
# prior for gamma (autocorrelation parameter) in state 1
gamma[2] ~ dbeta(1.5, 5) ## gamma for state 2
dev ~ dbeta(1,1) ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[1] <- gamma[2] + dev ## gamma for state 1
#Monthly Covaraites
for(x in 1:Months){
beta[x,1]~dnorm(beta_mu[1],beta_tau[1])
beta[x,2]<-0
beta2[x,1]~dnorm(beta2_mu[1],beta2_tau[1])
beta2[x,2]<-0
}
##Behavioral States
#Hierarchical structure across motnhs
#Intercepts
alpha_mu[1] ~ dnorm(0,0.386)
alpha_mu[2] ~ dnorm(0,0.386)
#Variance
alpha_tau[1] ~ dt(0,1,1)I(0,)
alpha_tau[2] ~ dt(0,1,1)I(0,)
#Slopes
## Ocean Depth
beta_mu[1] ~ dnorm(0,0.386)
beta_mu[2] = 0
# Distance coast
beta2_mu[1] ~ dnorm(0,0.386)
beta2_mu[2] = 0
#Monthly Variance
#Ocean
beta_tau[1] ~ dt(0,1,1)I(0,)
beta_tau[2] = 0
#Coast
beta2_tau[1] ~ dt(0,1,1)I(0,)
beta2_tau[2] = 0
#Probability of behavior switching
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]
##Argos priors##
#longitudinal argos error
argos_sigma ~ dunif(0,10)
#latitidunal argos error
argos_alpha~dunif(0,10)
#correlation in argos error
rho ~ dunif(-1, 1)
}"
,fill=TRUE)sink()
## user system elapsed
## 1.85 0.24 302.842.5 Chains
## Type Size PrettySize Rows Columns
## jagM rjags.parallel 29711288 [1] "28.3 Mb" 6 NA
## mdat data.frame 26066656 [1] "24.9 Mb" 57230 57
## m ggmap 13116432 [1] "12.5 Mb" 1280 1280
## d SpatialPointsDataFrame 7933544 [1] "7.6 Mb" 8130 60
## oxy data.frame 3443496 [1] "3.3 Mb" 7942 64
## sxy list 1994704 [1] "1.9 Mb" 21 NA
## mxy grouped_df 1871176 [1] "1.8 Mb" 3803 69
## data list 1128208 [1] "1.1 Mb" 11 NA
## argos array 691176 [1] "675 Kb" 2 12
## obs array 691176 [1] "675 Kb" 2 12## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1496363 80.0 2637877 140.9 2637877 140.9
## Vcells 12484315 95.3 39320772 300.0 56648713 432.2## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1491440 79.7 2637877 140.9 2637877 140.9
## Vcells 9601044 73.3 31456617 240.0 56648713 432.2
2.5.1 Compare to priors
2.6 Parameter Summary
## parameter par mean lower upper
## 1 alpha_mu alpha_mu[1] -0.96186842 -3.50549371 1.63657003
## 2 alpha_mu alpha_mu[2] 0.59121895 -1.79674752 3.24489361
## 3 beta beta[1,1] 0.21538466 -2.85255567 2.88795455
## 4 beta beta[2,1] 0.53873503 -2.47186522 3.37309464
## 5 beta beta[3,1] 0.45485619 -2.19903357 3.35863224
## 6 beta beta[4,1] 0.55599632 -2.24198234 3.48079918
## 7 beta beta[5,1] 0.16276653 -2.80347699 3.29991559
## 8 beta beta[1,2] 0.00000000 0.00000000 0.00000000
## 9 beta beta[2,2] 0.00000000 0.00000000 0.00000000
## 10 beta beta[3,2] 0.00000000 0.00000000 0.00000000
## 11 beta beta[4,2] 0.00000000 0.00000000 0.00000000
## 12 beta beta[5,2] 0.00000000 0.00000000 0.00000000
## 13 beta_mu beta_mu[1] 0.38788331 -1.76786224 2.55801932
## 14 beta_mu beta_mu[2] 0.00000000 0.00000000 0.00000000
## 15 beta2 beta2[1,1] -1.85955323 -4.37682736 -0.22298533
## 16 beta2 beta2[2,1] -1.78439248 -4.17171631 -0.49847004
## 17 beta2 beta2[3,1] -1.85084633 -4.18411487 -0.42928758
## 18 beta2 beta2[4,1] -1.73403601 -4.17343944 -0.24105829
## 19 beta2 beta2[5,1] -1.07793301 -4.25671057 0.30968408
## 20 beta2 beta2[1,2] 0.00000000 0.00000000 0.00000000
## 21 beta2 beta2[2,2] 0.00000000 0.00000000 0.00000000
## 22 beta2 beta2[3,2] 0.00000000 0.00000000 0.00000000
## 23 beta2 beta2[4,2] 0.00000000 0.00000000 0.00000000
## 24 beta2 beta2[5,2] 0.00000000 0.00000000 0.00000000
## 25 beta2_mu beta2_mu[1] -1.50654379 -3.75695277 -0.28775758
## 26 beta2_mu beta2_mu[2] 0.00000000 0.00000000 0.00000000
## 27 gamma gamma[1] 1.06718093 0.78679965 1.41921474
## 28 gamma gamma[2] 0.37542745 0.19924033 0.53489391
## 29 theta theta[1] -0.00444458 -0.08423803 0.09554119
## 30 theta theta[2] 3.29740006 0.34551004 6.17409858
3 Behavior and environment
3.1 Hierarchical
3.1.1 Ocean Depth
3.1.2 Distance to Coast
3.1.3 Interaction
3.2 By Month
3.2.1 Depth
Just the probability of feeding when traveling.
Just mean estimate.
3.2.2 Coast
Zooming in on the top right plot. 
Just mean estimate.
4 Behavioral Prediction
4.1 Spatial Prediction
4.1.1 Per Animal
## $`1`
##
## $`2`
4.2 Log Odds of Foraging
4.2.1 Ocean Depth
4.2.2 Distance From Coast
4.3 Autocorrelation in behavior
4.4 Behavioral description
4.5 Location of Behavior
5 Environmental Prediction - Probability of Foraging across time
5.1 Bathymetry
5.2 Distance to coast
5.3 All variables
6 Overlap with Krill Fishery
6.1 By Month
6.2 Change in foraging areas
Jan verus May
Red = Better Foraging in Jan Blue = Better Foraging in May
6.2.1 Variance in monthly suitability
6.2.2 Mean suitability
6.3 Monthly Overlap with Krill Fishery
## Type Size PrettySize Rows Columns
## pc tbl_df 30309904 [1] "28.9 Mb" 540000 12
## mdat data.frame 26066656 [1] "24.9 Mb" 57230 57
## temp ggmap 13116048 [1] "12.5 Mb" 1280 1280
## d SpatialPointsDataFrame 7933544 [1] "7.6 Mb" 8130 60
## oxy data.frame 3443496 [1] "3.3 Mb" 7942 64
## sxy list 1693456 [1] "1.6 Mb" 2 NA
## mxy data.frame 1669496 [1] "1.6 Mb" 3526 70
## allplot grouped_df 1031880 [1] "1007.7 Kb" 19780 7
## monthras list 858576 [1] "838.5 Kb" 5 NA
## mdf data.frame 650704 [1] "635.5 Kb" 9515 11## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1621821 86.7 5289763 282.6 16143078 862.2
## Vcells 20087661 153.3 120833612 921.9 271028039 2067.8